Theorem gagrp 19365

Description: The left argument of a group action is a group. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 30-Apr-2015.)

Assertion

Ref	Expression
gagrp	⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)

Proof of Theorem gagrp

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2770	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2770	. . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		eqid 2770	. . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺)
4	1, 2, 3	isga 19364	. . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( ⊕ :((Base‘𝐺) × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (((0g‘𝐺) ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))))
5	4	simplbi 501	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → (𝐺 ∈ Grp ∧ 𝑌 ∈ V))
6	5	simpld 499	1 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2150  ∀wral 3086  Vcvv 3462   × cxp 5663  ⟶wf 6536  ‘cfv 6540  (class class class)co 7414  Basecbs 17272  +_gcplusg 17313  0_gc0g 17495  Grpcgrp 19003   GrpAct cga 19362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8829  df-ga 19363

This theorem is referenced by:  gafo  19369  gass  19374  galcan  19377  gacan  19378  gapm  19379  gaorber  19381  gastacl  19382  galactghm  19477  sylow2alem2  19691  fxpsubm  33462  fxpsubg  33463  fxpsubrg  33464